## a very occasional diary.

 nondescript

## 2018-03-09

### Cographs, cocounts and coconuts.

Abstract: Dual of the familiar construction of the graph of a function is considered. The symmetry between graphs and cographs can be extended to a suprising degree.
Given a function $$f : A \rightarrow B$$, the graph of f  is defined as $$f^* = \{(x, f(x)) \mid x \in A\}.$$ In fact, within ZFC framework, functions are defined as graphs. A graph is a subset of the Cartesian product $$A \times B$$. One might want to associate to $$f$$ a dual cograph object: a certain quotient set of the disjoint sum $$A \sqcup B$$, which would uniquely identify the function. To understand the structure of the cograph, define the graph of a morphism $$f : A \rightarrow B$$ in a category with suitable products as a fibred product:
$$\require{AMScd}$$\begin{CD}
f^* @>\pi_2>> B\\
@V \pi_1 V V @VV 1_B V\\
A @>>f> B
\end{CD}In the category of sets this gives the standard definition. The cograph can be defined by a dual construction as a push-out:
\begin{CD}
A @>1_A>> A\\
@V f V V @VV j_1 V\\
B @>>j_2> f_*
\end{CD}Expanding this in the category of sets gives the following definition:
$$f_* = (A \sqcup B) / \pi_f,$$
where $$\pi_f$$ is the reflexive transitive closure of a relation $$\theta_f$$ given by (assuming in the following, without loss of generality, that $$A$$ and $$B$$ are disjoint)
$$x\overset{\theta_f}{\sim} y \equiv y = f(x)$$
That is, $$A$$ is partitioned by $$\pi_f$$ into subsets which are inverse images of elements of $$B$$ and to each such subset the element of $$B$$ which is its image is glued. This is somewhat similar to the mapping cylinder construction in topology. Some similarities between graphs and cographs are immediately obvious. For graphs: $$\forall x\in A\; \exists! y\in B\; (x, y)\in f^*$$ $$f(x) = \pi_2((\{x\}\times B) \cap f^*)$$ $$f(U) = \{y \mid y = f(x) \wedge x\in U \} = \pi_2((U\times B)\cap f^*)$$ where $$x\in A$$ and $$U\subseteq A$$. Similarly, for cographs: $$\forall x\in A\; \exists! y\in B\; [x] = [y]$$ $$f(x) = [x] \cap B$$ $$f(U) = (\bigcup [U])\cap B$$ where $$[x]$$ is the equivalance set of $$x$$ w.r.t. $$\pi_f$$ and $$[U] = \{[x] \mid x \in U\}$$. For inverse images: $$f^{-1}(y) = \pi_1((A \times \{y\}) \cap f^*) = [y] \cap A$$ $$f^{-1}(S) = \pi_1((A \times S) \cap f^*) = (\bigcup [S])\cap A$$ where $$y\in B$$ and $$S\subseteq B$$.

A graph can be expressed as $$f^* = \bigcup_{x \in A}(x, f(x))$$ To write out a similar representation of a cograph, we have to recall some elementary facts about equivalence relations.

Given a set $$A$$, let $$Eq(A) \subseteq Rel(A) = P(A \times A)$$ be the set of equivalence relations on $$A$$. For a relation $$\pi \subseteq A \times A$$, we have $$\pi \in Eq(A) \;\; \equiv \;\; \pi^0 = \Delta \subseteq \pi \; \wedge \; \pi^n = \pi, n \in \mathbb{Z}, n \neq 0.$$ To each $$\pi$$ corresponds a surjection $$A \twoheadrightarrow A/\pi$$. Assuming axiom of choice (in the form "all epics split"), an endomorphism $$A \twoheadrightarrow A/\pi \rightarrowtail A$$ can be assigned (non-uniquely) to $$\pi$$. It is easy to check, that this gives $$Eq(A) = End(A) / Aut(A)$$, where $$End(A)$$ and $$Aut(A)$$ are the monoid and the group of set endomorphisms and automorphisms respectively, with composition as the operation ($$Aut(A)$$ is not, in general, normal in $$End(A)$$, so $$Eq(A)$$ does not inherit any useful operation from this quotient set representation.). In addition to the monoid structure (w.r.t. composition) that $$Eq(A)$$ inherits from $$Rel(A)$$, it is also a lattice with infimum and supremum given by $$\pi \wedge \rho = \pi \cap \rho$$ $$\pi \vee \rho = \mathtt{tr}(\pi \cup \rho) = \bigcup_{n \in \mathbb{N}}(\pi \cup \rho)^n$$ For a subset $$X\subseteq A$$ define an equivalence relation $$e(X) = \Delta_A \cup (X\times X)$$, so that $$x\overset{e(X)}{\sim} y \equiv x = y \vee \{x, y\} \subseteq X$$ (Intuitively, $$e(X)$$ collapses $$X$$ to one point.) It is easy to check that $$f_* = \bigvee_{x \in A}e(\{x, f(x)\})$$ which is the desired dual of the graph representation above.